Felix Klein (1883) and Henri Poincaré (1882) conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves.
Henri Poincaré (1883) extended this to arbitrary multivalued analytic functions and gave informal arguments in its favor.
The history is described in Gray (1994); a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in de Saint-Gervais (2016) (the Bourbaki-type pseudonym of the group of fifteen mathematicians who jointly produced this publication).
Each such is conformally equivalent to a unique closed 2-manifold of constant curvature, so a quotient of one of the following by a free action of a discrete subgroup of an isometry group: The first case gives the 2-sphere, the unique 2-manifold with constant positive curvature and hence positive Euler characteristic (equal to 2).
In the context of closed Riemannian 2-manifolds, several modern proofs invoke nonlinear differential equations on the space of conformally equivalent metrics.
In 1913 Hermann Weyl published his classic textbook "Die Idee der Riemannschen Fläche" based on his Göttingen lectures from 1911 to 1912.
It was the first book to present the theory of Riemann surfaces in a modern setting and through its three editions has remained influential.
Springer (1957) followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem.
Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere (or equivalently if every Jordan curve separates it), then it is conformally equivalent to an open subset of the complex sphere.
Not every 3-manifold admits a geometry, but Thurston's geometrization conjecture proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable.